4472:, there is a unique conic passing through them. If three of these points lie on a line, then the conic is reducible, and may or may not be unique. If no four points are collinear, then five points define a unique conic (degenerate if three points are collinear, but the other two points determine the unique other line). If four points are collinear, however, then there is not a unique conic passing through them – one line passing through the four points, and the remaining line passes through the other point, but the angle is undefined, leaving 1 parameter free. If all five points are collinear, then the remaining line is free, which leaves 2 parameters free.
81:
132:
25:
175:
3068:
1462:
338:
1633:
4475:
Given four points in general linear position (no three collinear; in particular, no two coincident), there are exactly three pairs of lines (degenerate conics) passing through them, which will in general be intersecting, unless the points form a
2273:
4438:
Another type of degeneration occurs for an ellipse when the sum of the distances to the foci is mandated to equal the interfocal distance; thus it has semi-minor axis equal to zero and has eccentricity equal to one. The result is a
1985:
1047:
Over the complex projective plane there are only two types of degenerate conics – two different lines, which necessarily intersect in one point, or one double line. Any degenerate conic may be transformed by a
341:
Pencils of circles: in the pencil of red circles, the only degenerate conic is the horizontal axis; the pencil of blue circles has three degenerate conics, the vertical axis and two circles of radius zero.
1354:
4487:
Given three points, if they are non-collinear, there are three pairs of parallel lines passing through them – choose two to define one line, and the third for the parallel line to pass through, by the
1142:
1434:
1222:
2178:
3049:
1716:
1920:
1551:
4251:
3202:
4010:
1821:
2524:
2402:
985:
4575:
4179:
3944:
836:
549:
3885:
2463:
1622:
4321:
4373:
3123:
2772:
4426:
3731:
3443:
2956:
636:
389:
223:
73:
4683:
4065:
3685:
3348:
729:
442:
4110:
3635:
3524:
3403:
3312:
2556:
2309:
676:
3588:
122:
4648:
3557:
2600:
2341:
4612:
3765:
3234:
2856:
2798:
2065:
1257:
912:
874:
587:
480:
167:
2911:
3477:
2189:
2020:
769:
3379:
3265:
2982:
2885:
2827:
1741:
1037:
1011:
281:
In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and the
4257:
goes to 0; but, because they have conjugate complex points at infinity which become a double point on degeneration, cannot degenerate to two intersecting lines.
3659:
2668:
2640:
2620:
3802:
by considering the pencil of conics through the four roots of the quartic, and identifying the three degenerate conics with the three roots of the
4261:
Degenerate conics can degenerate further to more special degenerate conics, as indicated by the dimensions of the spaces and points at infinity.
1928:
4650:
corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of
2278:
The conic is degenerate if and only if the determinant of this matrix equals zero. In this case, we have the following possibilities:
3824:
In the complex projective plane, all conics are equivalent, and can degenerate to either two different lines or one double line.
2994:, four points leave one parameter free), of which three are degenerate, each consisting of a pair of lines, corresponding to the
1268:
1062:
4865:
3071:
1361:
1149:
4745:
2073:
4808:
2022:. This determinant is positive, zero, or negative as the conic is, respectively, an ellipse, a parabola, or a hyperbola.
321:
form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose
4765:
Faucette, William Mark (January 1996), "A Geometric
Interpretation of the Solution of the General Quartic Polynomial",
2997:
4883:
4799:
4767:
3799:
2690:
of the cone or when the cone degenerates to a cylinder and the plane is parallel to the axis of the cylinder. See
2678:
Conics, also known as conic sections to emphasize their three-dimensional geometry, arise as the intersection of a
1639:
1834:
1468:
4184:
4116:
goes to 0; but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
2707:
3128:
4465:
3949:
2991:
2646:. However, in other contexts it is not considered as a degenerate conic, as its equation is not of degree 2.
2025:
Analogously, a conic can be classified as non-degenerate or degenerate according to the discriminant of the
1746:
2468:
2346:
920:
444:, and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case
4513:
4452:
4122:
3890:
3809:
777:
493:
3834:
2407:
1556:
4268:
2711:
260:
4326:
1049:
296:
of conics. That is, if two real non-degenerated conics are defined by quadratic polynomial equations
4781:
3087:
2728:
4892:
4385:
3690:
3410:
595:
348:
182:
32:
4653:
4022:
3664:
3318:
2916:
681:
394:
4914:
4070:
3593:
3482:
3384:
3270:
3052:
2987:
2722:
2532:
2285:
641:
267:
3564:
4776:
4265:
Two intersecting lines can degenerate to two parallel lines, by rotating until parallel, as in
1262:
736:
286:
88:
4617:
3530:
2691:
2564:
2317:
2268:{\displaystyle Q={\begin{bmatrix}A&B&D\\B&C&E\\D&E&F\\\end{bmatrix}}.}
1055:
Over the real affine plane the situation is more complicated. A degenerate real conic may be:
4584:
3741:
3210:
2832:
2777:
2032:
1450:
1229:
879:
841:
554:
447:
139:
2890:
3450:
2718:
1993:
742:
483:
293:
3355:
3241:
2961:
2861:
2803:
1828:
1827:
Non-degenerate real conics can be classified as ellipses, parabolas, or hyperbolas by the
8:
3813:
2703:
1720:
1625:
1016:
990:
289:), or the null set (twice the line at infinity or two parallel complex conjugate lines).
252:
248:
4854:
4841:
4812:
4786:
4578:
4488:
3644:
2653:
2625:
2605:
2282:
Two intersecting lines (a hyperbola degenerated to its two asymptotes) if and only if
4879:
4861:
4382:
Two parallel lines can degenerate to a double line by moving into each other, as in
4469:
4444:
2679:
282:
271:
3791:), this is a Type I linear system of conics, and is animated in the linked video.
4443:(degenerate because the ellipse is not differentiable at the endpoints) with its
3803:
2687:
275:
2642:
are not both zero. This case always occurs as a degenerate conic in a pencil of
4751:
732:
256:
2650:
The case of coincident lines occurs if and only if the rank of the 3×3 matrix
589:
is an example of a degenerate conic consisting of twice the line at infinity.
4908:
4481:
3831:
Hyperbolas can degenerate to two intersecting lines (the asymptotes), as in
4440:
3775:
Note that this parametrization has a symmetry, where inverting the sign of
1442:
278:, a conic is degenerate if the plane goes through the vertex of the cones.
2706:
generally, arise as limits of non-degenerate conics, and are important in
3204:
yielding the following pencil; in all cases the center is at the origin:
329:
so that the given degenerate conic belongs to the pencil they determine.
244:
4845:
4816:
4790:
4494:
Given two distinct points, there is a unique double line through them.
4477:
4323:
or to a double line by rotating into each other about a point, as in
1461:
487:
337:
266:
Using the alternative definition of the conic as the intersection in
3051:
ways of choosing 2 pairs of points from 4 points (counting via the
2314:
Two parallel straight lines (a degenerate parabola) if and only if
1980:{\displaystyle M={\begin{bmatrix}A&B\\B&C\\\end{bmatrix}},}
232:
2673:
1632:
174:
131:
80:
24:
2990:(no three on a line), there is a pencil of conics through them (
2643:
2183:
the discriminant of this form is the determinant of the matrix
255:. This means that the defining equation is factorable over the
1358:
Two parallel complex conjugate lines (no real point), such as
1013:
into two lines, the line at infinity and the line of equation
4448:
2913:– throughout, one axis has length 2 and the other has length
240:
2683:
1449:
For any two degenerate conics of the same class, there are
1349:{\displaystyle x^{2}+y^{2}=0\Leftrightarrow (x+iy)(x-iy)=0}
4832:
Milne, J. J. (January 1926), "Note on
Degenerate Conics",
3125:
the pencil of conics through them can be parameterized as
1137:{\displaystyle x^{2}-y^{2}=0\Leftrightarrow (x+y)(x-y)=0}
2561:
A single line (and the line at infinity) if and only if
4797:
Lasley, Jr., J. W. (May 1957), "On
Degenerate Conics",
4432:
goes to 0, but cannot degenerate to non-parallel lines.
3001:
2919:
2204:
1943:
4656:
4620:
4587:
4516:
4388:
4329:
4271:
4187:
4125:
4073:
4025:
3952:
3893:
3837:
3744:
3693:
3667:
3647:
3596:
3567:
3533:
3485:
3453:
3413:
3387:
3358:
3321:
3273:
3244:
3213:
3131:
3090:
3000:
2986:
Such families arise naturally – given four points in
2964:
2893:
2864:
2835:
2806:
2780:
2731:
2656:
2628:
2608:
2567:
2535:
2529:
A single point (a degenerate ellipse) if and only if
2471:
2410:
2349:
2320:
2288:
2192:
2076:
2035:
1996:
1931:
1837:
1749:
1723:
1642:
1559:
1471:
1429:{\displaystyle x^{2}+1=0\Leftrightarrow (x+i)(x-i)=0}
1364:
1271:
1232:
1217:{\displaystyle x^{2}-1=0\Leftrightarrow (x+1)(x-1)=0}
1152:
1065:
1019:
993:
923:
882:
844:
780:
745:
684:
644:
598:
557:
496:
450:
397:
351:
185:
142:
91:
35:
2173:{\displaystyle Ax^{2}+2Bxy+Cy^{2}+2Dxz+2Eyz+Fz^{2};}
4435:
A double line cannot degenerate to the other types.
2670:is 1; in all other degenerate cases its rank is 2.
638:, which has only one real point, is degenerate, as
4853:
4677:
4642:
4606:
4569:
4420:
4367:
4315:
4245:
4173:
4104:
4059:
4004:
3938:
3879:
3759:
3725:
3679:
3653:
3629:
3582:
3551:
3518:
3471:
3437:
3397:
3373:
3342:
3306:
3259:
3228:
3196:
3117:
3043:
2976:
2950:
2905:
2879:
2850:
2821:
2792:
2766:
2662:
2634:
2614:
2594:
2550:
2518:
2457:
2396:
2335:
2303:
2267:
2172:
2059:
2014:
1979:
1914:
1815:
1735:
1710:
1616:
1545:
1428:
1348:
1251:
1216:
1136:
1052:into any other degenerate conic of the same type.
1031:
1005:
979:
906:
868:
830:
763:
723:
670:
630:
581:
543:
474:
436:
383:
217:
161:
116:
67:
1441:Twice the line at infinity (no real point in the
4906:
4019:Parabolas can degenerate to two parallel lines:
3044:{\displaystyle \textstyle {{\binom {4}{2,2}}=3}}
2686:. Degeneracy occurs when the plane contains the
2536:
2321:
2289:
391:is degenerate as its equation can be written as
4119:Ellipses can degenerate to two parallel lines:
3794:A striking application of such a family is in (
2674:Relation to intersection of a plane and a cone
4897:CRC Standard Mathematical Tables and Formulas
4827:, New York: The Macmillan Co., pp. x+405
3027:
3006:
1711:{\displaystyle 9x^{2}+12xy+4y^{2}-54x-36y+72}
1915:{\displaystyle Ax^{2}+2Bxy+Cy^{2}+2Dx+2Ey+F}
1546:{\displaystyle 3x^{2}-2xy-y^{2}-6x+10y-9=0,}
592:Similarly, the conic section with equation
263:) as the product of two linear polynomials.
4851:
4796:
4731:
4705:
1453:mapping the first conic to the second one.
4840:(180), The Mathematical Association: 7–9,
4780:
4246:{\displaystyle x^{2}+a^{2}y^{2}-a^{2}=0,}
3816:, when a conic degenerates to two lines.
2067:. Here the affine form is homogenized to
1922:, which is the determinant of the matrix
1226:A double line (multiplicity 2), such as
739:that intersect in the unique real point,
292:All these degenerate conics may occur in
16:2nd-degree plane curve which is reducible
4764:
3800:geometric solution to a quartic equation
3795:
3197:{\displaystyle (1+a)x^{2}+(1-a)y^{2}=2,}
2465:, and non-existent in the real plane if
1631:
1460:
336:
4005:{\displaystyle x^{2}-a^{2}y^{2}=a^{2},}
2343:. These lines are distinct and real if
4907:
4510:A simpler parametrization is given by
3445:ellipses with a horizontal major axis;
2702:Degenerate conics, as with degenerate
1823:is the union of the red and blue loci.
1816:{\displaystyle (3x+2y-6)(3x+2y-12)=0,}
1438:A single line and the line at infinity
4873:
4831:
4718:
2519:{\displaystyle D^{2}+E^{2}<(A+C)F}
2397:{\displaystyle D^{2}+E^{2}>(A+C)F}
980:{\displaystyle a(x^{2}+y^{2}-1)-bx=0}
4893:"7.2 The General Quadratic Equation"
4822:
4570:{\displaystyle ax^{2}+(1-a)y^{2}=1,}
4174:{\displaystyle x^{2}+a^{2}y^{2}-1=0}
3788:
3350:ellipses with a vertical major axis;
2404:(see second diagram), coincident if
1990:the matrix of the quadratic form in
774:The pencil of ellipses of equations
4809:Mathematical Association of America
4743:
4459:
3939:{\displaystyle x^{2}-a^{2}y^{2}=1,}
3084:For example, given the four points
3075:
917:The pencil of circles of equations
876:, into two parallel lines and, for
831:{\displaystyle ax^{2}+b(y^{2}-1)=0}
544:{\displaystyle a(x^{2}-y^{2})-b=0.}
251:of degree two) that fails to be an
13:
4669:
3880:{\displaystyle x^{2}-y^{2}=a^{2},}
3674:
3574:
3236:hyperbolae opening left and right;
3010:
2458:{\displaystyle D^{2}+E^{2}=(A+C)F}
1617:{\displaystyle (x-y+1)(3x+y-9)=0,}
14:
4926:
4825:Projective and related geometries
4800:The American Mathematical Monthly
4768:The American Mathematical Monthly
4316:{\displaystyle x^{2}-ay^{2}-1=0,}
2829:concretely, it is an ellipse for
1042:
735:. The conic consists thus of two
3066:
1059:Two intersecting lines, such as
345:The conic section with equation
285:), a single point (in fact, two
173:
130:
79:
23:
4368:{\displaystyle x^{2}-ay^{2}=0,}
3819:
3559:hyperbolae opening up and down,
2697:
1456:
1265:(only one real point), such as
4852:Pettofrezzo, Anthony (1978) ,
4724:
4711:
4698:
4545:
4533:
4504:
3671:
3479:the parallel horizontal lines
3172:
3160:
3144:
3132:
3118:{\displaystyle (\pm 1,\pm 1),}
3109:
3091:
2939:
2931:
2767:{\displaystyle x^{2}+ay^{2}=1}
2692:Conic section#Degenerate cases
2510:
2498:
2449:
2437:
2388:
2376:
2054:
2036:
2009:
1997:
1801:
1777:
1774:
1750:
1602:
1581:
1578:
1560:
1417:
1405:
1402:
1390:
1387:
1337:
1322:
1319:
1304:
1301:
1205:
1193:
1190:
1178:
1175:
1125:
1113:
1110:
1098:
1095:
959:
927:
819:
800:
758:
746:
718:
703:
700:
685:
526:
500:
425:
413:
410:
398:
1:
4691:
4421:{\displaystyle x^{2}-a^{2}=0}
3726:{\displaystyle x^{2}-y^{2}=0}
3438:{\displaystyle -1<a<0:}
2992:five points determine a conic
2951:{\textstyle 1/{\sqrt {|a|}},}
631:{\displaystyle x^{2}+y^{2}=0}
384:{\displaystyle x^{2}-y^{2}=0}
218:{\displaystyle x^{2}+y^{2}=0}
68:{\displaystyle x^{2}-y^{2}=0}
4856:Matrices and Transformations
4678:{\displaystyle 0,1,\infty .}
4480:(one pair is parallel) or a
4060:{\displaystyle x^{2}-ay-1=0}
3680:{\displaystyle a\to \infty }
3343:{\displaystyle 0<a<1:}
3267:the parallel vertical lines
1831:of the non-homogeneous form
1146:Two parallel lines, such as
724:{\displaystyle (x+iy)(x-iy)}
437:{\displaystyle (x-y)(x+y)=0}
7:
4105:{\displaystyle x^{2}-ay=0,}
3630:{\displaystyle y=x,\ y=-x;}
3519:{\displaystyle y=-1,\ y=1;}
3398:{\displaystyle {\sqrt {2}}}
3307:{\displaystyle x=-1,\ x=1;}
2551:{\displaystyle \det M>0}
2304:{\displaystyle \det M<0}
671:{\displaystyle x^{2}+y^{2}}
332:
259:(or more generally over an
10:
4931:
4484:(two pairs are parallel).
4453:radial elliptic trajectory
3887:or to two parallel lines:
3827:In the real affine plane:
3738:This then loops around to
3583:{\displaystyle a=\infty :}
310:, the conics of equations
261:algebraically closed field
4447:at the endpoints. As an
3787:. In the terminology of (
3065:
3060:
2721:of curves (1-dimensional
1628:of the red and blue loci.
1465:The degenerate hyperbola
1050:projective transformation
117:{\displaystyle x^{2}-1=0}
4834:The Mathematical Gazette
4747:Linear Systems of Conics
4643:{\displaystyle y^{2}=1,}
4497:
3810:Pappus's hexagon theorem
3661:and taking the limit as
3552:{\displaystyle a<-1:}
2595:{\displaystyle A=B=C=0,}
2336:{\displaystyle \det M=0}
1636:The degenerate parabola
4607:{\displaystyle x^{2}=1}
4468:: given five points in
3812:is the special case of
3760:{\displaystyle a>1,}
3229:{\displaystyle a>1:}
3053:multinomial coefficient
2988:general linear position
2858:two parallel lines for
2851:{\displaystyle a>0,}
2793:{\displaystyle a\neq 0}
2723:linear system of conics
2712:moduli spaces of curves
2060:{\displaystyle (x,y,z)}
1263:complex conjugate lines
1252:{\displaystyle x^{2}=0}
907:{\displaystyle a=1,b=0}
869:{\displaystyle a=0,b=1}
737:complex conjugate lines
582:{\displaystyle a=0,b=1}
475:{\displaystyle a=1,b=0}
287:complex conjugate lines
268:three-dimensional space
162:{\displaystyle x^{2}=0}
4874:Spain, Barry (2007) ,
4679:
4644:
4608:
4571:
4466:defined by five points
4422:
4369:
4317:
4247:
4175:
4106:
4061:
4006:
3946:or to the double line
3940:
3881:
3761:
3727:
3681:
3655:
3631:
3584:
3553:
3520:
3473:
3439:
3399:
3381:a circle (with radius
3375:
3344:
3308:
3261:
3230:
3198:
3119:
3045:
2978:
2958:which is infinity for
2952:
2907:
2906:{\displaystyle a<0}
2881:
2852:
2823:
2800:but is degenerate for
2794:
2774:is non-degenerate for
2768:
2664:
2636:
2616:
2596:
2552:
2520:
2459:
2398:
2337:
2305:
2269:
2174:
2061:
2016:
1981:
1916:
1824:
1817:
1737:
1712:
1629:
1618:
1547:
1451:affine transformations
1430:
1350:
1253:
1218:
1138:
1033:
1007:
981:
914:, into a double line.
908:
870:
832:
765:
725:
672:
632:
583:
545:
476:
438:
385:
342:
219:
163:
118:
69:
4680:
4645:
4609:
4572:
4423:
4370:
4318:
4248:
4176:
4107:
4062:
4007:
3941:
3882:
3762:
3728:
3682:
3656:
3632:
3585:
3554:
3521:
3474:
3472:{\displaystyle a=-1:}
3440:
3400:
3376:
3345:
3309:
3262:
3231:
3199:
3120:
3046:
2979:
2953:
2908:
2887:and a hyperbola with
2882:
2853:
2824:
2795:
2769:
2665:
2637:
2617:
2597:
2553:
2521:
2460:
2399:
2338:
2306:
2270:
2175:
2062:
2017:
2015:{\displaystyle (x,y)}
1982:
1917:
1818:
1738:
1713:
1635:
1619:
1548:
1464:
1431:
1351:
1254:
1219:
1139:
1034:
1008:
982:
909:
871:
833:
766:
764:{\displaystyle (0,0)}
726:
673:
633:
584:
546:
477:
439:
386:
340:
220:
164:
119:
70:
4823:Levy, Harry (1964),
4654:
4618:
4585:
4514:
4386:
4327:
4269:
4185:
4123:
4071:
4023:
3950:
3891:
3835:
3767:since pencils are a
3742:
3691:
3665:
3645:
3594:
3565:
3531:
3483:
3451:
3411:
3385:
3374:{\displaystyle a=0:}
3356:
3319:
3271:
3260:{\displaystyle a=1:}
3242:
3211:
3129:
3088:
2998:
2977:{\displaystyle a=0.}
2962:
2917:
2891:
2880:{\displaystyle a=0,}
2862:
2833:
2822:{\displaystyle a=0;}
2804:
2778:
2729:
2654:
2626:
2606:
2565:
2533:
2469:
2408:
2347:
2318:
2311:(see first diagram).
2286:
2190:
2074:
2033:
1994:
1929:
1835:
1747:
1721:
1640:
1557:
1469:
1362:
1269:
1230:
1150:
1063:
1017:
991:
921:
880:
842:
778:
743:
682:
642:
596:
555:
494:
448:
395:
349:
183:
140:
89:
33:
4579:affine combinations
4464:A general conic is
4181:or the double line
4067:or the double line
3590:the diagonal lines
2704:algebraic varieties
1736:{\displaystyle =0,}
1032:{\displaystyle x=0}
1006:{\displaystyle a=0}
249:polynomial equation
4675:
4640:
4604:
4567:
4489:parallel postulate
4418:
4365:
4313:
4243:
4171:
4102:
4057:
4002:
3936:
3877:
3757:
3723:
3677:
3651:
3627:
3580:
3549:
3516:
3469:
3435:
3395:
3371:
3340:
3304:
3257:
3226:
3194:
3115:
3041:
3040:
2974:
2948:
2903:
2877:
2848:
2819:
2790:
2764:
2660:
2632:
2612:
2592:
2548:
2516:
2455:
2394:
2333:
2301:
2265:
2256:
2170:
2057:
2029:quadratic form in
2012:
1977:
1968:
1912:
1825:
1813:
1733:
1708:
1630:
1614:
1543:
1426:
1346:
1249:
1214:
1134:
1029:
1003:
977:
904:
866:
828:
761:
721:
668:
628:
579:
551:The limiting case
541:
472:
434:
381:
343:
215:
159:
114:
65:
4876:Analytical Conics
4867:978-0-486-63634-4
4581:of the equations
3654:{\displaystyle a}
3611:
3503:
3393:
3291:
3082:
3081:
3025:
2943:
2717:For example, the
2663:{\displaystyle Q}
2635:{\displaystyle E}
2615:{\displaystyle D}
1743:which factors as
1553:which factors as
1261:Two intersecting
838:degenerates, for
678:is factorable as
253:irreducible curve
243:(a second-degree
4922:
4900:
4888:
4870:
4859:
4848:
4828:
4819:
4793:
4784:
4761:
4760:
4759:
4750:, archived from
4735:
4732:Pettofrezzo 1978
4728:
4722:
4715:
4709:
4706:Lasley, Jr. 1957
4702:
4685:
4684:
4682:
4681:
4676:
4649:
4647:
4646:
4641:
4630:
4629:
4613:
4611:
4610:
4605:
4597:
4596:
4576:
4574:
4573:
4568:
4557:
4556:
4529:
4528:
4508:
4470:general position
4460:Points to define
4427:
4425:
4424:
4419:
4411:
4410:
4398:
4397:
4375:in each case as
4374:
4372:
4371:
4366:
4355:
4354:
4339:
4338:
4322:
4320:
4319:
4314:
4297:
4296:
4281:
4280:
4252:
4250:
4249:
4244:
4233:
4232:
4220:
4219:
4210:
4209:
4197:
4196:
4180:
4178:
4177:
4172:
4158:
4157:
4148:
4147:
4135:
4134:
4111:
4109:
4108:
4103:
4083:
4082:
4066:
4064:
4063:
4058:
4035:
4034:
4011:
4009:
4008:
4003:
3998:
3997:
3985:
3984:
3975:
3974:
3962:
3961:
3945:
3943:
3942:
3937:
3926:
3925:
3916:
3915:
3903:
3902:
3886:
3884:
3883:
3878:
3873:
3872:
3860:
3859:
3847:
3846:
3814:Pascal's theorem
3798:) which gives a
3766:
3764:
3763:
3758:
3732:
3730:
3729:
3724:
3716:
3715:
3703:
3702:
3686:
3684:
3683:
3678:
3660:
3658:
3657:
3652:
3636:
3634:
3633:
3628:
3609:
3589:
3587:
3586:
3581:
3558:
3556:
3555:
3550:
3525:
3523:
3522:
3517:
3501:
3478:
3476:
3475:
3470:
3444:
3442:
3441:
3436:
3404:
3402:
3401:
3396:
3394:
3389:
3380:
3378:
3377:
3372:
3349:
3347:
3346:
3341:
3313:
3311:
3310:
3305:
3289:
3266:
3264:
3263:
3258:
3235:
3233:
3232:
3227:
3203:
3201:
3200:
3195:
3184:
3183:
3156:
3155:
3124:
3122:
3121:
3116:
3074:linear system, (
3070:
3069:
3058:
3057:
3050:
3048:
3047:
3042:
3039:
3032:
3031:
3030:
3024:
3009:
2983:
2981:
2980:
2975:
2957:
2955:
2954:
2949:
2944:
2942:
2934:
2929:
2927:
2912:
2910:
2909:
2904:
2886:
2884:
2883:
2878:
2857:
2855:
2854:
2849:
2828:
2826:
2825:
2820:
2799:
2797:
2796:
2791:
2773:
2771:
2770:
2765:
2757:
2756:
2741:
2740:
2708:compactification
2669:
2667:
2666:
2661:
2641:
2639:
2638:
2633:
2621:
2619:
2618:
2613:
2601:
2599:
2598:
2593:
2557:
2555:
2554:
2549:
2525:
2523:
2522:
2517:
2494:
2493:
2481:
2480:
2464:
2462:
2461:
2456:
2433:
2432:
2420:
2419:
2403:
2401:
2400:
2395:
2372:
2371:
2359:
2358:
2342:
2340:
2339:
2334:
2310:
2308:
2307:
2302:
2274:
2272:
2271:
2266:
2261:
2260:
2179:
2177:
2176:
2171:
2166:
2165:
2120:
2119:
2089:
2088:
2066:
2064:
2063:
2058:
2021:
2019:
2018:
2013:
1986:
1984:
1983:
1978:
1973:
1972:
1921:
1919:
1918:
1913:
1881:
1880:
1850:
1849:
1822:
1820:
1819:
1814:
1742:
1740:
1739:
1734:
1717:
1715:
1714:
1709:
1683:
1682:
1655:
1654:
1623:
1621:
1620:
1615:
1552:
1550:
1549:
1544:
1509:
1508:
1484:
1483:
1435:
1433:
1432:
1427:
1374:
1373:
1355:
1353:
1352:
1347:
1294:
1293:
1281:
1280:
1258:
1256:
1255:
1250:
1242:
1241:
1223:
1221:
1220:
1215:
1162:
1161:
1143:
1141:
1140:
1135:
1088:
1087:
1075:
1074:
1038:
1036:
1035:
1030:
1012:
1010:
1009:
1004:
987:degenerates for
986:
984:
983:
978:
952:
951:
939:
938:
913:
911:
910:
905:
875:
873:
872:
867:
837:
835:
834:
829:
812:
811:
793:
792:
771:, of the conic.
770:
768:
767:
762:
730:
728:
727:
722:
677:
675:
674:
669:
667:
666:
654:
653:
637:
635:
634:
629:
621:
620:
608:
607:
588:
586:
585:
580:
550:
548:
547:
542:
525:
524:
512:
511:
481:
479:
478:
473:
443:
441:
440:
435:
390:
388:
387:
382:
374:
373:
361:
360:
328:
324:
320:
309:
302:
283:line at infinity
237:degenerate conic
224:
222:
221:
216:
208:
207:
195:
194:
177:
168:
166:
165:
160:
152:
151:
134:
123:
121:
120:
115:
101:
100:
83:
74:
72:
71:
66:
58:
57:
45:
44:
27:
4930:
4929:
4925:
4924:
4923:
4921:
4920:
4919:
4905:
4904:
4903:
4899:(30th ed.)
4891:
4886:
4868:
4782:10.1.1.111.5574
4757:
4755:
4744:Coffman, Adam,
4739:
4738:
4729:
4725:
4716:
4712:
4703:
4699:
4694:
4689:
4688:
4655:
4652:
4651:
4625:
4621:
4619:
4616:
4615:
4592:
4588:
4586:
4583:
4582:
4552:
4548:
4524:
4520:
4515:
4512:
4511:
4509:
4505:
4500:
4462:
4406:
4402:
4393:
4389:
4387:
4384:
4383:
4350:
4346:
4334:
4330:
4328:
4325:
4324:
4292:
4288:
4276:
4272:
4270:
4267:
4266:
4228:
4224:
4215:
4211:
4205:
4201:
4192:
4188:
4186:
4183:
4182:
4153:
4149:
4143:
4139:
4130:
4126:
4124:
4121:
4120:
4078:
4074:
4072:
4069:
4068:
4030:
4026:
4024:
4021:
4020:
3993:
3989:
3980:
3976:
3970:
3966:
3957:
3953:
3951:
3948:
3947:
3921:
3917:
3911:
3907:
3898:
3894:
3892:
3889:
3888:
3868:
3864:
3855:
3851:
3842:
3838:
3836:
3833:
3832:
3822:
3804:resolvent cubic
3743:
3740:
3739:
3711:
3707:
3698:
3694:
3692:
3689:
3688:
3666:
3663:
3662:
3646:
3643:
3642:
3595:
3592:
3591:
3566:
3563:
3562:
3532:
3529:
3528:
3484:
3481:
3480:
3452:
3449:
3448:
3412:
3409:
3408:
3388:
3386:
3383:
3382:
3357:
3354:
3353:
3320:
3317:
3316:
3272:
3269:
3268:
3243:
3240:
3239:
3212:
3209:
3208:
3179:
3175:
3151:
3147:
3130:
3127:
3126:
3089:
3086:
3085:
3067:
3061:External videos
3026:
3014:
3005:
3004:
3003:
3002:
2999:
2996:
2995:
2963:
2960:
2959:
2938:
2930:
2928:
2923:
2918:
2915:
2914:
2892:
2889:
2888:
2863:
2860:
2859:
2834:
2831:
2830:
2805:
2802:
2801:
2779:
2776:
2775:
2752:
2748:
2736:
2732:
2730:
2727:
2726:
2700:
2676:
2655:
2652:
2651:
2627:
2624:
2623:
2607:
2604:
2603:
2566:
2563:
2562:
2534:
2531:
2530:
2489:
2485:
2476:
2472:
2470:
2467:
2466:
2428:
2424:
2415:
2411:
2409:
2406:
2405:
2367:
2363:
2354:
2350:
2348:
2345:
2344:
2319:
2316:
2315:
2287:
2284:
2283:
2255:
2254:
2249:
2244:
2238:
2237:
2232:
2227:
2221:
2220:
2215:
2210:
2200:
2199:
2191:
2188:
2187:
2161:
2157:
2115:
2111:
2084:
2080:
2075:
2072:
2071:
2034:
2031:
2030:
1995:
1992:
1991:
1967:
1966:
1961:
1955:
1954:
1949:
1939:
1938:
1930:
1927:
1926:
1876:
1872:
1845:
1841:
1836:
1833:
1832:
1748:
1745:
1744:
1722:
1719:
1718:
1678:
1674:
1650:
1646:
1641:
1638:
1637:
1558:
1555:
1554:
1504:
1500:
1479:
1475:
1470:
1467:
1466:
1459:
1369:
1365:
1363:
1360:
1359:
1289:
1285:
1276:
1272:
1270:
1267:
1266:
1237:
1233:
1231:
1228:
1227:
1157:
1153:
1151:
1148:
1147:
1083:
1079:
1070:
1066:
1064:
1061:
1060:
1045:
1018:
1015:
1014:
992:
989:
988:
947:
943:
934:
930:
922:
919:
918:
881:
878:
877:
843:
840:
839:
807:
803:
788:
784:
779:
776:
775:
744:
741:
740:
733:complex numbers
683:
680:
679:
662:
658:
649:
645:
643:
640:
639:
616:
612:
603:
599:
597:
594:
593:
556:
553:
552:
520:
516:
507:
503:
495:
492:
491:
449:
446:
445:
396:
393:
392:
369:
365:
356:
352:
350:
347:
346:
335:
326:
322:
311:
304:
297:
257:complex numbers
247:, defined by a
229:
228:
227:
226:
225:
203:
199:
190:
186:
184:
181:
180:
178:
170:
169:
147:
143:
141:
138:
137:
135:
126:
125:
124:
96:
92:
90:
87:
86:
84:
76:
75:
53:
49:
40:
36:
34:
31:
30:
28:
17:
12:
11:
5:
4928:
4918:
4917:
4915:Conic sections
4902:
4901:
4889:
4884:
4871:
4866:
4849:
4829:
4820:
4794:
4762:
4740:
4737:
4736:
4723:
4710:
4696:
4695:
4693:
4690:
4687:
4686:
4674:
4671:
4668:
4665:
4662:
4659:
4639:
4636:
4633:
4628:
4624:
4603:
4600:
4595:
4591:
4577:which are the
4566:
4563:
4560:
4555:
4551:
4547:
4544:
4541:
4538:
4535:
4532:
4527:
4523:
4519:
4502:
4501:
4499:
4496:
4461:
4458:
4457:
4456:
4436:
4433:
4417:
4414:
4409:
4405:
4401:
4396:
4392:
4380:
4364:
4361:
4358:
4353:
4349:
4345:
4342:
4337:
4333:
4312:
4309:
4306:
4303:
4300:
4295:
4291:
4287:
4284:
4279:
4275:
4259:
4258:
4242:
4239:
4236:
4231:
4227:
4223:
4218:
4214:
4208:
4204:
4200:
4195:
4191:
4170:
4167:
4164:
4161:
4156:
4152:
4146:
4142:
4138:
4133:
4129:
4117:
4101:
4098:
4095:
4092:
4089:
4086:
4081:
4077:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4033:
4029:
4017:
4001:
3996:
3992:
3988:
3983:
3979:
3973:
3969:
3965:
3960:
3956:
3935:
3932:
3929:
3924:
3920:
3914:
3910:
3906:
3901:
3897:
3876:
3871:
3867:
3863:
3858:
3854:
3850:
3845:
3841:
3821:
3818:
3773:
3772:
3756:
3753:
3750:
3747:
3735:
3734:
3722:
3719:
3714:
3710:
3706:
3701:
3697:
3676:
3673:
3670:
3650:
3638:
3637:
3626:
3623:
3620:
3617:
3614:
3608:
3605:
3602:
3599:
3579:
3576:
3573:
3570:
3560:
3548:
3545:
3542:
3539:
3536:
3526:
3515:
3512:
3509:
3506:
3500:
3497:
3494:
3491:
3488:
3468:
3465:
3462:
3459:
3456:
3446:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3406:
3392:
3370:
3367:
3364:
3361:
3351:
3339:
3336:
3333:
3330:
3327:
3324:
3314:
3303:
3300:
3297:
3294:
3288:
3285:
3282:
3279:
3276:
3256:
3253:
3250:
3247:
3237:
3225:
3222:
3219:
3216:
3193:
3190:
3187:
3182:
3178:
3174:
3171:
3168:
3165:
3162:
3159:
3154:
3150:
3146:
3143:
3140:
3137:
3134:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3080:
3079:
3063:
3062:
3038:
3035:
3029:
3023:
3020:
3017:
3013:
3008:
2973:
2970:
2967:
2947:
2941:
2937:
2933:
2926:
2922:
2902:
2899:
2896:
2876:
2873:
2870:
2867:
2847:
2844:
2841:
2838:
2818:
2815:
2812:
2809:
2789:
2786:
2783:
2763:
2760:
2755:
2751:
2747:
2744:
2739:
2735:
2699:
2696:
2675:
2672:
2659:
2648:
2647:
2631:
2611:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2559:
2547:
2544:
2541:
2538:
2527:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2492:
2488:
2484:
2479:
2475:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2431:
2427:
2423:
2418:
2414:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2370:
2366:
2362:
2357:
2353:
2332:
2329:
2326:
2323:
2312:
2300:
2297:
2294:
2291:
2276:
2275:
2264:
2259:
2253:
2250:
2248:
2245:
2243:
2240:
2239:
2236:
2233:
2231:
2228:
2226:
2223:
2222:
2219:
2216:
2214:
2211:
2209:
2206:
2205:
2203:
2198:
2195:
2181:
2180:
2169:
2164:
2160:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2118:
2114:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2087:
2083:
2079:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2011:
2008:
2005:
2002:
1999:
1988:
1987:
1976:
1971:
1965:
1962:
1960:
1957:
1956:
1953:
1950:
1948:
1945:
1944:
1942:
1937:
1934:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1879:
1875:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1848:
1844:
1840:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1732:
1729:
1726:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1681:
1677:
1673:
1670:
1667:
1664:
1661:
1658:
1653:
1649:
1645:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1507:
1503:
1499:
1496:
1493:
1490:
1487:
1482:
1478:
1474:
1458:
1455:
1447:
1446:
1439:
1436:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1372:
1368:
1356:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1292:
1288:
1284:
1279:
1275:
1259:
1248:
1245:
1240:
1236:
1224:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1160:
1156:
1144:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1086:
1082:
1078:
1073:
1069:
1044:
1043:Classification
1041:
1028:
1025:
1022:
1002:
999:
996:
976:
973:
970:
967:
964:
961:
958:
955:
950:
946:
942:
937:
933:
929:
926:
903:
900:
897:
894:
891:
888:
885:
865:
862:
859:
856:
853:
850:
847:
827:
824:
821:
818:
815:
810:
806:
802:
799:
796:
791:
787:
783:
760:
757:
754:
751:
748:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
690:
687:
665:
661:
657:
652:
648:
627:
624:
619:
615:
611:
606:
602:
578:
575:
572:
569:
566:
563:
560:
540:
537:
534:
531:
528:
523:
519:
515:
510:
506:
502:
499:
471:
468:
465:
462:
459:
456:
453:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
400:
380:
377:
372:
368:
364:
359:
355:
334:
331:
214:
211:
206:
202:
198:
193:
189:
179:
172:
171:
158:
155:
150:
146:
136:
129:
128:
127:
113:
110:
107:
104:
99:
95:
85:
78:
77:
64:
61:
56:
52:
48:
43:
39:
29:
22:
21:
20:
19:
18:
15:
9:
6:
4:
3:
2:
4927:
4916:
4913:
4912:
4910:
4898:
4894:
4890:
4887:
4885:0-486-45773-7
4881:
4877:
4872:
4869:
4863:
4858:
4857:
4850:
4847:
4843:
4839:
4835:
4830:
4826:
4821:
4818:
4814:
4810:
4806:
4802:
4801:
4795:
4792:
4788:
4783:
4778:
4774:
4770:
4769:
4763:
4754:on 2018-07-02
4753:
4749:
4748:
4742:
4741:
4733:
4727:
4720:
4714:
4707:
4701:
4697:
4672:
4666:
4663:
4660:
4657:
4637:
4634:
4631:
4626:
4622:
4601:
4598:
4593:
4589:
4580:
4564:
4561:
4558:
4553:
4549:
4542:
4539:
4536:
4530:
4525:
4521:
4517:
4507:
4503:
4495:
4492:
4490:
4485:
4483:
4482:parallelogram
4479:
4473:
4471:
4467:
4454:
4450:
4446:
4442:
4437:
4434:
4431:
4415:
4412:
4407:
4403:
4399:
4394:
4390:
4381:
4378:
4362:
4359:
4356:
4351:
4347:
4343:
4340:
4335:
4331:
4310:
4307:
4304:
4301:
4298:
4293:
4289:
4285:
4282:
4277:
4273:
4264:
4263:
4262:
4256:
4240:
4237:
4234:
4229:
4225:
4221:
4216:
4212:
4206:
4202:
4198:
4193:
4189:
4168:
4165:
4162:
4159:
4154:
4150:
4144:
4140:
4136:
4131:
4127:
4118:
4115:
4099:
4096:
4093:
4090:
4087:
4084:
4079:
4075:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4031:
4027:
4018:
4015:
3999:
3994:
3990:
3986:
3981:
3977:
3971:
3967:
3963:
3958:
3954:
3933:
3930:
3927:
3922:
3918:
3912:
3908:
3904:
3899:
3895:
3874:
3869:
3865:
3861:
3856:
3852:
3848:
3843:
3839:
3830:
3829:
3828:
3825:
3817:
3815:
3811:
3807:
3805:
3801:
3797:
3796:Faucette 1996
3792:
3790:
3786:
3782:
3778:
3770:
3754:
3751:
3748:
3745:
3737:
3736:
3720:
3717:
3712:
3708:
3704:
3699:
3695:
3668:
3648:
3641:(dividing by
3640:
3639:
3624:
3621:
3618:
3615:
3612:
3606:
3603:
3600:
3597:
3577:
3571:
3568:
3561:
3546:
3543:
3540:
3537:
3534:
3527:
3513:
3510:
3507:
3504:
3498:
3495:
3492:
3489:
3486:
3466:
3463:
3460:
3457:
3454:
3447:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3407:
3390:
3368:
3365:
3362:
3359:
3352:
3337:
3334:
3331:
3328:
3325:
3322:
3315:
3301:
3298:
3295:
3292:
3286:
3283:
3280:
3277:
3274:
3254:
3251:
3248:
3245:
3238:
3223:
3220:
3217:
3214:
3207:
3206:
3205:
3191:
3188:
3185:
3180:
3176:
3169:
3166:
3163:
3157:
3152:
3148:
3141:
3138:
3135:
3112:
3106:
3103:
3100:
3097:
3094:
3077:
3073:
3064:
3059:
3056:
3054:
3036:
3033:
3021:
3018:
3015:
3011:
2993:
2989:
2984:
2971:
2968:
2965:
2945:
2935:
2924:
2920:
2900:
2897:
2894:
2874:
2871:
2868:
2865:
2845:
2842:
2839:
2836:
2816:
2813:
2810:
2807:
2787:
2784:
2781:
2761:
2758:
2753:
2749:
2745:
2742:
2737:
2733:
2725:) defined by
2724:
2720:
2715:
2713:
2709:
2705:
2695:
2694:for details.
2693:
2689:
2685:
2681:
2671:
2657:
2645:
2629:
2609:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2560:
2545:
2542:
2539:
2528:
2513:
2507:
2504:
2501:
2495:
2490:
2486:
2482:
2477:
2473:
2452:
2446:
2443:
2440:
2434:
2429:
2425:
2421:
2416:
2412:
2391:
2385:
2382:
2379:
2373:
2368:
2364:
2360:
2355:
2351:
2330:
2327:
2324:
2313:
2298:
2295:
2292:
2281:
2280:
2279:
2262:
2257:
2251:
2246:
2241:
2234:
2229:
2224:
2217:
2212:
2207:
2201:
2196:
2193:
2186:
2185:
2184:
2167:
2162:
2158:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2116:
2112:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2085:
2081:
2077:
2070:
2069:
2068:
2051:
2048:
2045:
2042:
2039:
2028:
2023:
2006:
2003:
2000:
1974:
1969:
1963:
1958:
1951:
1946:
1940:
1935:
1932:
1925:
1924:
1923:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1877:
1873:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1846:
1842:
1838:
1830:
1810:
1807:
1804:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1730:
1727:
1724:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1679:
1675:
1671:
1668:
1665:
1662:
1659:
1656:
1651:
1647:
1643:
1634:
1627:
1611:
1608:
1605:
1599:
1596:
1593:
1590:
1587:
1584:
1575:
1572:
1569:
1566:
1563:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1505:
1501:
1497:
1494:
1491:
1488:
1485:
1480:
1476:
1472:
1463:
1454:
1452:
1444:
1440:
1437:
1423:
1420:
1414:
1411:
1408:
1399:
1396:
1393:
1384:
1381:
1378:
1375:
1370:
1366:
1357:
1343:
1340:
1334:
1331:
1328:
1325:
1316:
1313:
1310:
1307:
1298:
1295:
1290:
1286:
1282:
1277:
1273:
1264:
1260:
1246:
1243:
1238:
1234:
1225:
1211:
1208:
1202:
1199:
1196:
1187:
1184:
1181:
1172:
1169:
1166:
1163:
1158:
1154:
1145:
1131:
1128:
1122:
1119:
1116:
1107:
1104:
1101:
1092:
1089:
1084:
1080:
1076:
1071:
1067:
1058:
1057:
1056:
1053:
1051:
1040:
1026:
1023:
1020:
1000:
997:
994:
974:
971:
968:
965:
962:
956:
953:
948:
944:
940:
935:
931:
924:
915:
901:
898:
895:
892:
889:
886:
883:
863:
860:
857:
854:
851:
848:
845:
825:
822:
816:
813:
808:
804:
797:
794:
789:
785:
781:
772:
755:
752:
749:
738:
734:
715:
712:
709:
706:
697:
694:
691:
688:
663:
659:
655:
650:
646:
625:
622:
617:
613:
609:
604:
600:
590:
576:
573:
570:
567:
564:
561:
558:
538:
535:
532:
529:
521:
517:
513:
508:
504:
497:
490:of equations
489:
485:
469:
466:
463:
460:
457:
454:
451:
431:
428:
422:
419:
416:
407:
404:
401:
378:
375:
370:
366:
362:
357:
353:
339:
330:
318:
314:
307:
300:
295:
290:
288:
284:
279:
277:
274:and a double
273:
269:
264:
262:
258:
254:
250:
246:
242:
238:
234:
212:
209:
204:
200:
196:
191:
187:
176:
156:
153:
148:
144:
133:
111:
108:
105:
102:
97:
93:
82:
62:
59:
54:
50:
46:
41:
37:
26:
4896:
4875:
4855:
4837:
4833:
4824:
4804:
4798:
4775:(1): 51–57,
4772:
4766:
4756:, retrieved
4752:the original
4746:
4726:
4713:
4700:
4506:
4493:
4486:
4474:
4463:
4451:, this is a
4441:line segment
4429:
4376:
4260:
4254:
4113:
4013:
3826:
3823:
3820:Degeneration
3808:
3793:
3784:
3780:
3776:
3774:
3768:
3083:
2985:
2716:
2701:
2698:Applications
2677:
2649:
2277:
2182:
2026:
2024:
1989:
1829:discriminant
1826:
1457:Discriminant
1448:
1443:affine plane
1054:
1046:
916:
773:
591:
344:
316:
312:
305:
298:
291:
280:
265:
236:
230:
4811:: 362–364,
2027:homogeneous
245:plane curve
4758:2013-07-03
4719:Spain 2007
4692:References
4379:goes to 0.
4016:goes to 0.
3769:projective
488:hyperbolas
4878:, Dover,
4860:, Dover,
4777:CiteSeerX
4670:∞
4540:−
4478:trapezoid
4400:−
4341:−
4299:−
4283:−
4222:−
4160:−
4085:−
4046:−
4037:−
3964:−
3905:−
3849:−
3789:Levy 1964
3779:reverses
3705:−
3675:∞
3672:→
3619:−
3575:∞
3541:−
3493:−
3461:−
3415:−
3281:−
3167:−
3104:±
3095:±
2785:≠
1796:−
1769:−
1694:−
1685:−
1597:−
1567:−
1529:−
1511:−
1498:−
1486:−
1412:−
1388:⇔
1329:−
1302:⇔
1200:−
1176:⇔
1164:−
1120:−
1096:⇔
1077:−
963:−
954:−
814:−
731:over the
710:−
530:−
514:−
405:−
363:−
103:−
47:−
4909:Category
333:Examples
233:geometry
4846:3602237
4817:2309606
4791:2975214
3687:yields
3076:Coffman
2682:with a
2644:circles
1624:is the
482:in the
294:pencils
4882:
4864:
4844:
4815:
4789:
4779:
3610:
3502:
3290:
3072:Type I
2719:pencil
484:pencil
4842:JSTOR
4813:JSTOR
4807:(5),
4787:JSTOR
4498:Notes
4449:orbit
3771:line.
2680:plane
1626:union
272:plane
270:of a
241:conic
239:is a
4880:ISBN
4862:ISBN
4614:and
4445:foci
3783:and
3749:>
3538:<
3427:<
3421:<
3332:<
3326:<
3218:>
2898:<
2840:>
2688:apex
2684:cone
2622:and
2602:and
2543:>
2496:<
2374:>
2296:<
325:and
303:and
276:cone
235:, a
4773:103
4428:as
4253:as
4112:as
4012:as
3055:).
2710:of
2537:det
2322:det
2290:det
486:of
319:= 0
308:= 0
301:= 0
231:In
4911::
4895:,
4838:13
4836:,
4805:64
4803:,
4785:,
4771:,
4491:.
3806:.
3405:);
3078:).
2972:0.
2714:.
1799:12
1706:72
1697:36
1688:54
1660:12
1523:10
1039:.
539:0.
317:bg
315:+
313:af
4734:)
4730:(
4721:)
4717:(
4708:)
4704:(
4673:.
4667:,
4664:1
4661:,
4658:0
4638:,
4635:1
4632:=
4627:2
4623:y
4602:1
4599:=
4594:2
4590:x
4565:,
4562:1
4559:=
4554:2
4550:y
4546:)
4543:a
4537:1
4534:(
4531:+
4526:2
4522:x
4518:a
4455:.
4430:a
4416:0
4413:=
4408:2
4404:a
4395:2
4391:x
4377:a
4363:,
4360:0
4357:=
4352:2
4348:y
4344:a
4336:2
4332:x
4311:,
4308:0
4305:=
4302:1
4294:2
4290:y
4286:a
4278:2
4274:x
4255:a
4241:,
4238:0
4235:=
4230:2
4226:a
4217:2
4213:y
4207:2
4203:a
4199:+
4194:2
4190:x
4169:0
4166:=
4163:1
4155:2
4151:y
4145:2
4141:a
4137:+
4132:2
4128:x
4114:a
4100:,
4097:0
4094:=
4091:y
4088:a
4080:2
4076:x
4055:0
4052:=
4049:1
4043:y
4040:a
4032:2
4028:x
4014:a
4000:,
3995:2
3991:a
3987:=
3982:2
3978:y
3972:2
3968:a
3959:2
3955:x
3934:,
3931:1
3928:=
3923:2
3919:y
3913:2
3909:a
3900:2
3896:x
3875:,
3870:2
3866:a
3862:=
3857:2
3853:y
3844:2
3840:x
3785:y
3781:x
3777:a
3755:,
3752:1
3746:a
3733:)
3721:0
3718:=
3713:2
3709:y
3700:2
3696:x
3669:a
3649:a
3625:;
3622:x
3616:=
3613:y
3607:,
3604:x
3601:=
3598:y
3578::
3572:=
3569:a
3547::
3544:1
3535:a
3514:;
3511:1
3508:=
3505:y
3499:,
3496:1
3490:=
3487:y
3467::
3464:1
3458:=
3455:a
3433::
3430:0
3424:a
3418:1
3391:2
3369::
3366:0
3363:=
3360:a
3338::
3335:1
3329:a
3323:0
3302:;
3299:1
3296:=
3293:x
3287:,
3284:1
3278:=
3275:x
3255::
3252:1
3249:=
3246:a
3224::
3221:1
3215:a
3192:,
3189:2
3186:=
3181:2
3177:y
3173:)
3170:a
3164:1
3161:(
3158:+
3153:2
3149:x
3145:)
3142:a
3139:+
3136:1
3133:(
3113:,
3110:)
3107:1
3101:,
3098:1
3092:(
3037:3
3034:=
3028:)
3022:2
3019:,
3016:2
3012:4
3007:(
2969:=
2966:a
2946:,
2940:|
2936:a
2932:|
2925:/
2921:1
2901:0
2895:a
2875:,
2872:0
2869:=
2866:a
2846:,
2843:0
2837:a
2817:;
2814:0
2811:=
2808:a
2788:0
2782:a
2762:1
2759:=
2754:2
2750:y
2746:a
2743:+
2738:2
2734:x
2658:Q
2630:E
2610:D
2590:,
2587:0
2584:=
2581:C
2578:=
2575:B
2572:=
2569:A
2558:.
2546:0
2540:M
2526:.
2514:F
2511:)
2508:C
2505:+
2502:A
2499:(
2491:2
2487:E
2483:+
2478:2
2474:D
2453:F
2450:)
2447:C
2444:+
2441:A
2438:(
2435:=
2430:2
2426:E
2422:+
2417:2
2413:D
2392:F
2389:)
2386:C
2383:+
2380:A
2377:(
2369:2
2365:E
2361:+
2356:2
2352:D
2331:0
2328:=
2325:M
2299:0
2293:M
2263:.
2258:]
2252:F
2247:E
2242:D
2235:E
2230:C
2225:B
2218:D
2213:B
2208:A
2202:[
2197:=
2194:Q
2168:;
2163:2
2159:z
2155:F
2152:+
2149:z
2146:y
2143:E
2140:2
2137:+
2134:z
2131:x
2128:D
2125:2
2122:+
2117:2
2113:y
2109:C
2106:+
2103:y
2100:x
2097:B
2094:2
2091:+
2086:2
2082:x
2078:A
2055:)
2052:z
2049:,
2046:y
2043:,
2040:x
2037:(
2010:)
2007:y
2004:,
2001:x
1998:(
1975:,
1970:]
1964:C
1959:B
1952:B
1947:A
1941:[
1936:=
1933:M
1910:F
1907:+
1904:y
1901:E
1898:2
1895:+
1892:x
1889:D
1886:2
1883:+
1878:2
1874:y
1870:C
1867:+
1864:y
1861:x
1858:B
1855:2
1852:+
1847:2
1843:x
1839:A
1811:,
1808:0
1805:=
1802:)
1793:y
1790:2
1787:+
1784:x
1781:3
1778:(
1775:)
1772:6
1766:y
1763:2
1760:+
1757:x
1754:3
1751:(
1731:,
1728:0
1725:=
1703:+
1700:y
1691:x
1680:2
1676:y
1672:4
1669:+
1666:y
1663:x
1657:+
1652:2
1648:x
1644:9
1612:,
1609:0
1606:=
1603:)
1600:9
1594:y
1591:+
1588:x
1585:3
1582:(
1579:)
1576:1
1573:+
1570:y
1564:x
1561:(
1541:,
1538:0
1535:=
1532:9
1526:y
1520:+
1517:x
1514:6
1506:2
1502:y
1495:y
1492:x
1489:2
1481:2
1477:x
1473:3
1445:)
1424:0
1421:=
1418:)
1415:i
1409:x
1406:(
1403:)
1400:i
1397:+
1394:x
1391:(
1385:0
1382:=
1379:1
1376:+
1371:2
1367:x
1344:0
1341:=
1338:)
1335:y
1332:i
1326:x
1323:(
1320:)
1317:y
1314:i
1311:+
1308:x
1305:(
1299:0
1296:=
1291:2
1287:y
1283:+
1278:2
1274:x
1247:0
1244:=
1239:2
1235:x
1212:0
1209:=
1206:)
1203:1
1197:x
1194:(
1191:)
1188:1
1185:+
1182:x
1179:(
1173:0
1170:=
1167:1
1159:2
1155:x
1132:0
1129:=
1126:)
1123:y
1117:x
1114:(
1111:)
1108:y
1105:+
1102:x
1099:(
1093:0
1090:=
1085:2
1081:y
1072:2
1068:x
1027:0
1024:=
1021:x
1001:0
998:=
995:a
975:0
972:=
969:x
966:b
960:)
957:1
949:2
945:y
941:+
936:2
932:x
928:(
925:a
902:0
899:=
896:b
893:,
890:1
887:=
884:a
864:1
861:=
858:b
855:,
852:0
849:=
846:a
826:0
823:=
820:)
817:1
809:2
805:y
801:(
798:b
795:+
790:2
786:x
782:a
759:)
756:0
753:,
750:0
747:(
719:)
716:y
713:i
707:x
704:(
701:)
698:y
695:i
692:+
689:x
686:(
664:2
660:y
656:+
651:2
647:x
626:0
623:=
618:2
614:y
610:+
605:2
601:x
577:1
574:=
571:b
568:,
565:0
562:=
559:a
536:=
533:b
527:)
522:2
518:y
509:2
505:x
501:(
498:a
470:0
467:=
464:b
461:,
458:1
455:=
452:a
432:0
429:=
426:)
423:y
420:+
417:x
414:(
411:)
408:y
402:x
399:(
379:0
376:=
371:2
367:y
358:2
354:x
327:g
323:f
306:g
299:f
213:0
210:=
205:2
201:y
197:+
192:2
188:x
157:0
154:=
149:2
145:x
112:0
109:=
106:1
98:2
94:x
63:0
60:=
55:2
51:y
42:2
38:x
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